When using the half-angle formulas for trigonometric functions of alpha/2, I determine the sign based on the quadrant in which alpha lies.Determine whether the statement makes sense or does not make sense, and explain your reasoning.

When using the half-angle formulas for trigonometric functions of alpha/2, I determine the sign based on the quadrant in which alpha lies.Determine whether the statement makes sense or does not make sense, and explain your reasoning.

Question
Trigonometric Functions
asked 2020-12-16
When using the half-angle formulas for trigonometric functions of \(\displaystyle\frac{\alpha}{{2}}\), I determine the sign based on the quadrant in which \(\displaystyle\alpha\) lies.Determine whether the statement makes sense or does not make sense, and explain your reasoning.

Answers (1)

2020-12-17
For example:
The trigonometric function is given as, \(\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}.\)
\(\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}={\cos{{\left(\frac{{225}^{\circ}}{{2}}\right)}}}\)
Here, \(\displaystyle\alpha={225}^{\circ}.\)
Apply the half angle formula for the given trigonometric function, \(\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}\) in which the angle \(\displaystyle{112.5}^{\circ}\) lies in the quadrant (II) where sine and cosecant is positive. We have to determine the value of \(\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}\) which is negative in second quadrant.
Therefore in half angle formula , negative sign has to be put.
\(\displaystyle{\cos{{\left(\frac{\alpha}{{2}}\right)}}}=-\sqrt{{\frac{{{1}+{\cos{\alpha}}}}{{2}}}}\)
The value of \(\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}={\cos{{\left(\frac{{225}^{\circ}}{{2}}\right)}}}{i}{s},\)
\(\displaystyle{\cos{{\left(\frac{{225}^{\circ}}{{2}}\right)}}}=-\sqrt{{\frac{{{1}+{\cos{{225}}}^{\circ}}}{{2}}}}\)
\(\displaystyle=\sqrt{{\frac{{{1}+{\left(-\frac{\sqrt{{2}}}{{2}}\right)}}}{{2}}}}\)
\(\displaystyle=-\sqrt{{{\left(\frac{{{2}-\sqrt{{2}}}}{{2}}\right)}}}\)
0

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